Tuesday

March 3, 2015

March 3, 2015

Posted by **Claire** on Tuesday, February 15, 2011 at 3:01am.

1- Find the corresponding particular solution (in implicit form) that satisfies the initial; condition y = when x = 0.

2- Find the explicit form of this particular solution.

3- What is the value of y given by this particular solution when x =1? Give your answer to four decimal places.

regards Claire

- differential equations -
**MathMate**, Tuesday, February 15, 2011 at 9:52amTake the example of a differential equation:

Since the differential equation in question has not been posted, I will work with an example.

y"-2y'+2y=e^{2x}.....(1)

The solution yh to the homogeneous equation

y"-2y'+2y=0 .... (2)

is

yh=C1*e^{x}+C2*x*e^{x}

Note the two integration constants.

To obtain the general solution to equation (1), we must add to yh, solution of (2) a particular solution that satisfies (1). This is called a particular solution, yp.

For this particular problem,

yp=x²e^{x}

which can be obtained by the method of variation of parameters.

Therefore the general solution of (1) is the sum of yh and yp, i.e.

y = C1*e^{x}+ C2*x*e^{x}+ x²e^{x}...(general solution)

However, y still contains the constants C1 and C2 to be determined. They can be found by the initial conditions, such as y(0)=0 and y'(0)=1.

Substitute (0,0) into the general solution, we obtain an equation in terms of C1 and C2.

Differentiate the general solution once and substitute x=0, y'(0)=1, we get another equation. From these equations, we can solve for C1 and C2 which is the solution to the problem.

The above general solution is explicit because y, and only y, appears only on the left hand side.

A solution of the form

xy²+x²=2y²

is called an implicit solution because y appears more than once in an equation.

**Answer this Question**

**Related Questions**

Maths - Hi I am really sorry I now people have already attempted to help me with...

Calculus-HELP PLZ !!!! - Find a general solution to the first – degree ODE. If ...

Calculus - Find a general solution to each first ¨C degree ordinary differential...

Calculus-HELP!!! - Find a general solution to the first – degree ordinary ...

Calculus - Find a general solution to each first – degree ODE. If an initial ...

calculus - Please help. I have an implicit answer to a differential equation of ...

Calculus - Find the particular solution that satisfies the intial condition: xdy...

calculus - Sorry to post this again, but I am still unable to understand it and ...

STEVE or REINY-plz help! - Find a general solution to the first – degree ...

Calculus - Using 3(x-3)(x^2-6x+23)^1/2, as the chain rule differentiation of f(x...